Question

Show that any positive odd integer is of the form $4q + 1$ or $4q + 3$ where q is the positive integer.

Answer 1

Hint:
 we can use the Euclid division lemma which states that, If a and b are two positive integers then we can write,
a=bq + r where $0 \leqslant r < b$
Now assume the positive integers to be a and b=$4$ and put it in the formula. Then put the value of r to find if the equation gives an odd positive integer or not.

Complete step by step solution:
We have to show that any positive odd integer is of form $4q + 1$ or $4q + 3$ where q is the positive integer.
On using Euclid’s division lemma, If a and b are two positive integers then we can write,
a=bq + r where $0 \leqslant r < b$
Then let the positive integers be a and b=$4$
Then we can write,
a=$4q + r$--- (i) where $0 \leqslant r < 4$
There here r can be either greater or equal to zero and less than $4$
So if r=$1$ then eq. (i) becomes-
$ \Rightarrow a = 4q + 1$
Now this will always be an odd integer whether we put any positive integer value for q.
Now if we put r=$3$ then eq. (i) becomes-
$ \Rightarrow a = 4q + 3$
Now this will always be an odd integer whether we put any positive integer value for q.

Hence Any positive odd integer is of form $4q + 1$ or $4q + 3$ where q is the positive integer, Proved.

Note:
Here we have not put r=$2$ because then we will get,
$ \Rightarrow a = 4q + 2$
And if we put any positive integer value in place of q we will get only positive even numbers.